C₂×#4312017 #72294763

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#4312017

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

61205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(20,21)(24,25);(7,8)(9,10)(14,18)(16,17)(19,20,23,21)(22,25,26,24);(4,6)(8,10)(12,13)(16,17)(19,20,23,21)(22,25,26,24);(3,4)(5,6)(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,21,23,20)(22,25,26,24);(11,14)(12,16)(13,17)(15,18)(19,22)(20,24)(21,25)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293914, C₂×#72231749, C₂×#72323417, …
maximal normal subgroup of	C₂×#72293914, C₂×#72231749, C₂×#72323417, …
maximal quotient of	C₂×#72293958, C₂×#72278705

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(20,21)(24,25), (7,8)(9,10)(14,18)(16,17)(19,20,23,21)(22,25,26,24), (4,6)(8,10)(12,13)(16,17)(19,20,23,21)(22,25,26,24), (3,4)(5,6)(7,8)(9,10)(11,13,15,12)(14,17,18,16)(19,21,23,20)(22,25,26,24), (11,14)(12,16)(13,17)(15,18)(19,22)(20,24)(21,25)(23,26), (1,2));

C2×#4312017 #72294763